1. Motion in One Dimension
Chapter 2
Motion in One Dimension

2. Dynamics
The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts
Kinematics is a part of dynamics
How things move: describing motion
Not concerned with why things move, the cause of the motion

3. Quantities in Motion Any motion involves three concepts
Displacement
Velocity
Acceleration
These concepts can be used to study objects in motion

4. Position Defined in terms of a frame of reference
One dimensional, so generally the x- or y-axis
Defines a starting point for the motion

5. Displacement Defined as the change in position
f stands for final and i stands for initial
May be represented as y if vertical
Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

6. Displacements

7. Vector and Scalar Quantities
Vector quantities need both magnitude (size) and direction to completely describe them
Generally denoted by boldfaced type and an arrow over the letter
+ or – sign is sufficient for this chapter
Scalar quantities are completely described by magnitude only

8. Displacement Isn’t Distance
The displacement of an object is not the same as the distance it travels
Example: Throw a ball straight up and then catch it at the same point you released it
The distance is twice the height
The displacement is zero

9. Speed
The average speed of an object is defined as the total distance traveled divided by the total time elapsed
Speed is a scalar quantity

10. Speed, cont
Average speed totally ignores any variations in the object’s actual motion during the trip
The total distance and the total time are all that is important
SI units are m/s

11. Velocity It takes time for an object to undergo a displacement
The average velocity is rate at which the displacement occurs
generally use a time interval, so let ti = 0

12. Velocity continued
Direction will be the same as the direction of the displacement (time interval is always positive)
+ or - is sufficient
Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.)
Other units may be given in a problem, but generally will need to be converted to these

13. Speed vs. Velocity
Cars on both paths have the same average velocity since they had the same displacement in the same time interval
The car on the blue path will have a greater average speed since the distance it traveled is larger

14. Graphical Interpretation of Velocity
Velocity can be determined from a position-time graph
Average velocity equals the slope of the line joining the initial and final positions
An object moving with a constant velocity will have a graph that is a straight line

15. Average Velocity, Constant
The straight line indicates constant velocity
The slope of the line is the value of the average velocity

16. Instantaneous Velocity
The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero
The instantaneous velocity indicates what is happening at every point of time

17. Acceleration We have investigated various types of accelerations….
Catapult of a/c carrier
Concussion
Drag race
rollercoaster
The following slides are technical details about the concepts and equations

18. Acceleration Changing velocity means an acceleration is present
Acceleration is the rate of change of the velocity
Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

19. Accelerations can be +, -
We won’t be using the word ‘decelerating’
We will be paying close attention to the sign of the acceleration
Can you determine an object’s motion from the sign of the velocity and of the acceleration?

20. Relationship Between Acceleration and Velocity
constant velocity (red arrows of same size, same direction)
Acceleration equals zero

21. Relationship Between Velocity and Acceleration
Velocity and acceleration are in the same direction
Acceleration is uniform
(blue arrows maintain the same length)
Velocity is increasing, speeding up
(red arrows are getting longer)
velocity and acceleration = same sign

22. Relationship Between Velocity and Acceleration
Acceleration and velocity are in opposite directions
Acceleration is uniform
(blue arrows maintain the same length)
Velocity is decreasing, slowing down
(red arrows are getting shorter)
Velocity, acceleration = opposite signs

23. Acceleration
What if you flipped direction of motion in the previous slides?
When speeding up (no matter what direction),
the sign of the velocity and the acceleration are the same
When slowing down (no matter what the direction),
the sign of the velocity and the acceleration are opposite

24. Graphical Interpretation of Acceleration
is the slope of the line connecting the initial and final velocities on a velocity-time graph

25. Graphical Interpretation of the Equation

26. Remember linear and quadratic equations ?
linear equation y = m x + b
y is variable on y-axis
x is variable on x-axis
M is the slope of the line
B is the y intercept
Quadratic equation y = a x2 + b x + c
a, b, c are coefficients

27. Compare to the equation for constant velocity…
d = v t + di
d is position on y-axis, t is time on x-axis
These are the “VARIABLES” in the equation
v is slope of the line, aka velocity
di is the initial position
There are 4 ‘numbers’
If you know any three, you can solve for the unknown one

28. Compare to equations for accelerated motion…
d = ½ a t2 + vi t + di
d is position on the y-axis
t is time on the x-axis
These are the “variables” in the equation
Acceleration a, initial velocity vi and initial position di are the ‘coefficients’
There are 6 ‘numbers’ in this equation: if you know 5, you can solve for the unknown one.

29. Summary: constant velocity
Position vs. time (linear)
d = v t + di
V is constant for the entire problem

30. Summary:accelerated motion
Position vs time (quadratic, parabola)
d = ½ a t2 + vi t + di
Velocity vs time (linear)
v = a t + vi
Acceleration is whatever is given or calculated
It is a specific unchanging number

31. Additional accel equations
V2 = vi2 + 2 a (∆ d)
A useful equation that doesn’t involve time

32. Problem-Solving Hints
First read the entire problem
Draw a diagram
Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations
Identify all the given information
Rewrite, be sure all the units are consistent
Convert if necessary
What are you solving for?
Choose the appropriate kinematic equation

33. Problem-Solving Hints, cont
Solve for the unknowns
One equation, only one number will remain unknown
Check your results
Does the sign (direction) of the answer make sense?
Does the size, magnitude of the answer make sense? Too big or small??